Graph structures and well-quasi-ordering

Robertson and Seymour proved that graphs are well-quasi-ordered by the
minor relation and the weak immersion relation. In other words, given
infinitely many graphs, one graph contains another as a minor (or a weak
immersion, respectively). Unlike the relation of minor and weak
immersion, the topological minor relation does not well-quasi-order
graphs in general. However, Robertson conjectured in the late 1980s
that for every positive integer k, the topological minor relation
well-quasi-orders graphs that do not contain a topological minor
isomorphic to the path of length k with each edge duplicated. We will
sketch the idea of our recent proof of this conjecture. In addition, we
will give a structure theorem for excluding a fixed graph as a
topological minor. Such structure theorems were previously obtained by
Grohe and Marx and by Dvorak, but we push one of the bounds in their
theorems to the optimal value. This improvement is needed for our proof
of Robertson's conjecture. This work is joint with Robin Thomas.